Is it permissible for a woman to travel a distance of a day and a night without a mahram, and what are the legal proofs for that? And is it permissible to compromise on some religious matters, such as the hijab, when inviting non-Muslims to Islam with the aim of facilitating their embracing it?

Calling people to Islam is one of the best deeds. The caller (Da'i) should have knowledge of what he is calling to and wisdom in his methods of invitation. He should proceed gradually with the person being called (Mad'u), starting with what is most important, then what is important, such as focusing on faith and monotheism first. It is permissible for the caller to temporarily remain silent about some reprehensible acts to prepare the opportune time to forbid them. However, it is not permissible to compromise any part of Islam or change any of its rulings. Among the most important means of Da'wah are seeking Allah's help, being kind to the person being called, and persistently supplicating for them.